Polynomial Structures for Nilpotent Groups
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 1, January 1996
POLYNOMIAL STRUCTURES FOR NILPOTENT GROUPS
KAREL DEKIMPE, PAUL IGODT, AND KYUNG BAI LEE
Abstract. If a polycyclic-by-finite rank-K group Γ admits a faithful affine K representation making it acting on R properly discontinuously and with com- pact quotient, we say that Γ admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups Γ. Very recently examples have been obtained showing that, even for torsion-free, finitely gen- erated nilpotent groups N, affine structures do not always exist. It looks natural to consider affine structures as examples of polynomial structures of degree one. We introduce the concept of a canonical type polynomial structure for polycyclic-by-finite groups. Using the algebraic framework of the Seifert Fiber Space construction and a nice cohomology vanishing theorem, we prove the existence and uniqueness (up to conjugation) of canonical type polyno- mial structures for virtually finitely generated nilpotent groups. Applying this uniqueness to a result going back to Mal0cev, it follows that, for torsion-free, finitely generated nilpotent groups, each canonical polynomial structure is ex- pressed in polynomials of limited degree. The minimal degree needed for ob- taining a polynomial structure will determine the “affine defect number”. We prove that the known counterexamples to Milnor’s question have the smallest possible affine defect, i.e. affine defect number equal to one.
1. Introduction and main result Let G be a connected and simply connected solvable Lie group. Milnor ([Mil77]) conjectured that G admits a complete affinely flat structure invariant under left translations. Since then, a few papers claiming this conjecture affirmatively have been published, for example, ([Boy89], [Boy90] and [Nis90]). With this in mind, the idea of a canonical type affine representation (implicitly born in [Sch74] and [Lee83]) for (virtually) torsion-free, finitely generated nilpotent groups and the iteration problem related to this have been investigated ([DI93]). Uniqueness of these representations was studied in ([IL90]). However, today, counterexamples produced by Benoist ([Ben92]) and Burde and Grunewald ([BG93]) show that certain nilmanifolds do not admit a complete affinely flat structure. Consequently, it follows that Milnor’s conjecture is false even in the nilpotent case. Note that one can look at such a complete affinely flat structure as a “polynomial structure of degree 1”. In a situation where “polynomial structures of degree 1” fail to exist, the next best structure will be “polynomial structure of higher degree”.
Received by the editors May 8, 1994. 1991 Mathematics Subject Classification. Primary 57S30, 22E40, 22E25 . Key words and phrases. Lattices in nilpotent Lie groups, complete affinely flat structures. The first author is Research Assistant of the National Fund For Scientific Research (Belgium).
c 1996 American Mathematical Society
77
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The purpose of this paper is to prove that every torsion-free, finitely generated nilpotent group of rank K admits a faithful “canonical type” representation into the group of polynomial diffeomorphisms of RK . This result will be achieved using the algebraic framework of the so-called Seifert Fiber Space construction. As a consequence of this approach, uniqueness of such canonical representations is also proved; moreover, existence and uniqueness stay valid for groups which are virtually nilpotent. Based on the uniqueness and a well-known canonical polynomial structure found already by Mal0cev, it looks interesting to introduce the notion of affine defect number for a torsion-free f.g. nilpotent group. Groups of affine defect 0 are exactly the fundamental groups of the complete affinely flat manifolds. Groups of affine defect > 0 are those obstructing Milnor’s question. In the closing section we prove that the examples produced by Benoist, Burde and Grunewald are of affine defect 1. More precisely, the 11-dimensional nilpotent Lie groups which fail to admit a complete affinely flat structure do admit a polynomial structure of degree 2. This is the lowest possible such a structure.
2. Canonical type structures It is well known ([Seg83, Lemma 6, p. 16]) that, if Γ is a polycyclic–by–finite group, then there exists an ascending sequence (or filtration) of normal subgroups Γi (0 i c +1)ofΓ ≤ ≤ Γ :Γ0=1 Γ1 Γ2 Γc 1 Γc Γc+1 =Γ ∗ ⊂ ⊂ ⊂···⊂ − ⊂ ⊂ for which k Γi/Γi 1 = Z i for 1 i c and some ki N0 and − ∼ ≤ ≤ ∈ Γ/Γc is finite.
Moreover, each Γi can be chosen to be a characteristic subgroup of Γ. Let us call such a filtration of Γ a torsion-free filtration.WeuseKto denote the Hirsch number (or rank) of Γ. Often, we will also use Ki = ki + ki+1 + + kc and ··· Kc+1 = 0. It follows that K = K1. Definition 2.1. Assume Γ is polycyclic–by–finite with a torsion-free filtration Γ . ∗ A set of generators A = a1,1,a1,2,... ,a1,k1 ,a2,1,... ,ac,kc ,α1,... ,αr of Γ will be called compatible with{ Γ iff } ∗ i 1,2,... ,c : a1,1,a1,2,... ,ai,k generates Γi. ∀ ∈{ } i It is clear at once that any torsion-free filtration Γ of Γ admits a compatible set of generators. ∗ In the sequel, frequently we will work with quotient groups of Γ. In order to avoid complicated notations, we will write the same symbols for elements in Γ, as for their coset classes in a quotient group. E.g. we say that Γ2/Γ1 is the free abelian
group generated by a2,1,a2,2,... ,a2,k2 . Now we introduce quickly the basic building blocks for the representations we are interested in (see [Lee83] and [CR77]):
(RK , Rk)= continuous maps λ : RK Rk , M { → } (RK )= homeomorphisms h : RK RK . H { → }
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(RK , Rk) is an abelian group (for the addition of maps) and can be made into a M GL(k, R) (RK)–module, if we define ×H (g,h) 1 K k K λ = g λ h− , λ (R ,R ), g GL(k, R), h (R ). ◦ ◦ ∀ ∈M ∀ ∈ ∀ ∈H We remark that there is an embedding (RK , Rk) (GL(k, R) (RK)) , M o ×H → (Rk+K )with H (1) (λ, g, h)(x, y)=(g(x)+λh(y),h(y)),
x Rk, y RK , λ (RK,Rk), g GL(k, R), h (RK). ∀ ∈ ∀ ∈ ∀ ∈M ∀ ∈ ∀ ∈H Definition 2.2. Assume Γ is polycyclic–by–finite with a torsion-free filtration Γ . K ∗ A representation ρ = ρ0 :Γ (R ) will be called of canonical type with respect to Γ iff it induces a sequence→H of representations: ∗ K ρi :Γ/Γi (R i+1 ), (1 i c) →H ≤ ≤ such that for all i the following diagram commutes: (2) k 1 Z i = Γi/Γi 1 Γ/Γi 1 Γ/Γi 1 → ∼ − → − → → j ρi 1 Bi ρi ↓ ↓ − ↓ × Ki+1 ki Ki+1 1 (R , R ) (G ) GL(ki, Z) (R ) 1 →M →Mo ×H → ×H → where
Ki+1 ki Ki+1 (G ) stands for (R , R ) (GL(ki, Z) (R )), •Mo ×H M o ×H j(z):RKi+1 Rki : x z, z Zki and • → 7→ ∀ ∈ k Bi :Γ/Γi GL(ki, Z) denotes the action of Γ/Γi induced on Z i by conju- • → gation in Γ/Γi 1. − Let us establish the following results, which will be of use in a characterisation of canonical type representations. Theorem 2.3. Let Γ be any group acting on a contractible space M and containing a normal subgroup Γ0 of finite index, which acts freely and properly discontinuously on M, such that the quotient space M =Γ0 Mis a closed asphericalf manifold. \ Then, the kernel of the induced action of F =Γ/Γ0 on M is the isomorphic image of thef torsion of CΓ(Γ0) inside F . In fact, the setf of all torsion of CΓ(Γ0) forms a characteristic subgroup of Γ, and equals the kernel of the action of Γ on M. Proof. Let t be a torsion element in C (Γ ). This means that in the commutative Γ 0 f diagram 1 Γ0 Γ F 1 → → → → ↓↓ ↓ 1Inn(Γ0) Aut(Γ0) Out(Γ0) 1 →→ → → the group T generated by t maps trivially into Aut(Γ0). (The vertical maps are induced by conjugation in Γ). By [LR81, Lemma 1] the image of T in F acts trivially on M. This image is isomorphic to T ,asΓ0 is torsion-free. Conversely, if T 0 is a subgroup of F acting trivially on M,thenT0 is the isomorphic image of a subgroup T of Γ (this lifting of T 0 back to Γ can be done already if the T 0 action on M has a fixed point). Clearly, then T commutes with Γ0 so that T CΓ(Γ0). ⊂ For the last statement, note that a torsion element of Γ acts effectively on M if and only if the corresponding element of F acts effectively on M. f
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As an immediate consequence, we find the following reformulation of the above theorem in the situation we’re interested in. Corollary 2.4. Let Γ be a polycyclic–by–finite group, with Hirsch number K.Let Nbe the Γc in a torsion-free filtration of Γ,andletF denote the set of torsion of K CΓ(N). For any properly discontinuous action of ρ :Γ R ,F is the kernel of ρ. Moreover, F is the maximal finite normal subgroup of →Γ. We are now ready to summarize some important results on (general) represen- tations of canonical type. For elements x RK we will label the coordinates as follows: ∈
x =(x1,1,x2,1,... ,x1,k1 ,x2,1,... ,x2,k2 ,x3,1,... ,xc,kc ).
We also write xi to indicate all variables (xi,1,xi,2,... ,xi,k ). If ρ = ρ0 :Γ i → (RK) is a representation, then for each γ Γ there exist continuous functions Hγ K ∈ hi,j : R R such that → K K γ γ γ ρ(γ):R R :x (h1,1(x),h1,2(x),... ,h (x)). → 7→ c,kc In a way similar as for the x we use hγ to indicate (hγ (x),hγ (x),... ,hγ (x)). i,j i i,1 i,2 i,ki Theorem 2.5. Let Γ be polycyclic–by–finite, Γ be a torsion-free filtration of Γ. ∗ Then, there exists a representation ρ :Γ (RK)which is of canonical type →H K with respect to Γ . Moreover, any two such representations ρ, ρ0 :Γ (R )are ∗ →H conjugate to each other inside (RK ). Also, for each canonical type representation H ρ :Γ (RK)we have: →H (1) Γ acts properly discontinuously on RK ,viaρand with compact quotient. (2) ker ρ =the maximal finite normal subgroup of Γ,soρis effective iff Γ is almost torsion-free (i.e. has no non-trivial finite normal subgroups)(see [DI94]). (3) γ Γ: ∀ ∈ K K γ γ γ ρ(γ):R R :x (h1 (x),h2(x),... ,hc(x)) is such that → 7→ γ γ hi (x)=Bi(γ)xi+gi(xi+1,xi+2,... ,xc),
γ Ki+1 ki for some continuous function gi : R R . K → k (4) Γi acts trivially on R i+1 and via affine transformations on R i . Proof. The proof of the existence and uniqueness of a canonical type representation can be found in [LR84]. The reader may also consult [LR84] to see that the action of Γ defined on RK is properly discontinuous and that the orbit space is compact. (2) follows from Corollary 2.4. (3) and (4) are immediate consequences of the iterative way of building up ρ (see also (1)).
Having this concept of a (general) canonical type representation, it is clear now that to get a nicer geometric structure one needs to use smaller subgroups of (RK ). In view of the iterative set up, it is however necessary that these subgroupsH satisfy one crucial condition: assume we restrict (RK , Rk) to a subspace (RK , Rk) M S containing the space of constant mappings Rk and assume we restrict (RK )to H a subgroup H(RK ); then one observes that it is necessary that (RK , Rk)isa S S (GL(k, R) H(RK))-submodule such that there is a monomorphism, (RK , Rk) ×S S o (GL(k, R) H(RK )) , H(Rk+K). Possible examples of such situations are ×S →S
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K k Smooth representations: Define C∞(R , R ) to denote the vector space of functions f : RK Rk, which are infinitely many times differentiable and de- K → note by C∞(R ) the group, under composition of maps, of smooth diffeomor- K K k K k phisms of R . Then by considering C∞(R , R ) instead of (R , R )and K K M replacing (R )byC∞(R ), we find the so–called canonical type smooth representations.H Here, results analogous to Theorem 2.5 can be formulated (see [CR77] and [LR84]). Affine representations: Restrict (RK , Rk)toAff(RK,Rk) (the vector space M of affine mappings) and (RK )toAff(RK)(theaffinegroupofRK)(see [DI93]). H Polynomial representations: Write P (RK , Rk ) to refer to the vector space of polynomial mappings p : RK Rk.Sopis given by k polynomials → in K variables. P (RK ) will be used to indicate the set of all polynomial diffeomorphisms of RK , with an inverse which is also a polynomial mapping. This is a group where the multiplication is given by composition. It is not hard to verify that P (RK , Rk)isaAut(Zk) P(RK)-module and that the resulting semi-direct product group × P (RK , Rk) (Aut Zk P (RK )) P (RK+k) o × ⊆ by defining (p, g, h) P (RK , Rk) (Aut Zk P (RK )), (x, y) Rk+K : ∀ ∈ o × ∀ ∈ (p,g,h)(x, y)=(gx + p(h(y)),h(y)). Restrict (RK , Rk)toP(RK,Rk)and (RK)toP(RK), and we will speak of canonicalM type polynomial representations.H In each of these restricted situations one now faces existence and uniqueness ques- tions.
K k Counterexample 2.6. It looks tempting to consider Ps(R , R ), the vector space of polynomial mappings p : RK Rk which are of total degree s.Sopis now given by k polynomials in K variables,→ each of which is of total≤ degree s.Use K ≤ K Pt(R )to indicate a group consisting only of polynomial diffeomorphisms of R of K k k K degree t. Now remark, however, that Ps(R , R ) will not be a Aut(Z ) Pt(R )- module,≤ as the degrees go up under the group-action. ×
3. Canonical type polynomial structures for nilpotent groups Assume,fromnowon,thatNis a torsion-free, finitely generated nilpotent group of rank K and class c. N is polycyclic and hence has torsion-free filtrations. Of particular interest in our context are filtrations which are central series of length c, e.g. the upper central series. Another frequently arising filtration of this kind is obtained as follows. Write γ1(N)=N and γi+1(N)=[N,γi(N)] for the terms of N the lower central series of N.Now,itiswellknownthat γk(N) [the so-called root set or isolator of γk(N), i.e. the set of elements having a power in γk(N)] N N p N is a characteristic subgroup of N,with[ γk(N), γl(N)] γk+l(N)and N N ⊆ N γk(N)/ γk+1(N) torsion-free. We now define Nc = N, Nc 1 = γ2(N), ..., p p − p N = N γ (N). p1 cp p So, the torsion-free filtrations we are looking at are central series of the form: p N : N0 =1 N1 N2 Nc 1 Nc =Nc+1 = N. ∗ ⊂ ⊂ ⊂···⊂ − ⊂ We will refer to such a filtration, as a torsion-free central series.
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From now on, if we speak of a polynomial representation ρ : N P (RK )which is of canonical type, we mean of canonical type with respect to some→ torsion-free central series. Referring to Theorem 2.5 and to the fact that we are dealing with a central series, we can state the following corollary. Corollary 3.1. Assume that N is as above and that N is some torsion-free central ∗ series. If ρ : N P (RK ) is of canonical type with respect to N ,then, n N: K K→ γ γ γ ∗ ∀ ∈ ρ(n):R R :x (p1 (x),p2(x),... ,pc(x)) is such that → 7→ γ n pi (x)=xi+Qi(xi+1,xi+2,... ,xc),
n Ki+1 ki for some polynomial mapping Qi : R R . Moreover, Ni acts trivially on → RKi+1 and as translations on the ith block Rki , in such a way that these translations generate all of Rki . Example 3.2. 1 2 Let N : a1,1,a1,2,a2,1,a2,2,a3,1,a3,2 [a3,2,a3,1]=a1−,1a2,1a2,3 . h k 2 i [a2,1,a3,1]=a1−,1 [a2,2,a3,2]=a1,1 2 [a2,3,a2,2]=a1,1 [a2,2,a2,1]=a1,1 all other commutators trivial. One can check that there is a canonical type representation ρ : N P ( R 6 )of this N given in terms of the images of the generators (the central series−→ used, is suggested by the labeling of the generators):
ρ(a1,1)(x)=(1+x1,1,x2,1,x2,2,x2,3,x3,1,x3,2), ρ(a2,1)(x)=(x1,1 2x3,1,1+x2,1,x2,2,x2,3,x3,1,x3,2), − ρ(a2,2)(x)=(x1,1+2x2,1 +x3,2,x2,1,1+x2,2,x2,3,x3,1,x3,2), ρ(a2,3)(x)=(x1,1+x2,2,x2,1,x2,2,1+x2,3,x3,1,x3,2), ρ(a3,1)(x)=(x1,1,x2,1,x2,2,x2,3,1+x3,1,x3,2), 2 ρ(a3,2)(x)=(x1,1+2x2,2x3,1 x ,x2,1 +x3,1,x2,2,x2,3 +2x3,1,x3,1,1+x3,2). − 3,1 The following theorem, intrinsically due to Mal0cev ([Mal51]), will imply the exis- tence of canonical type polynomial representations for all torsion-free, f.g. nilpotent groups. Theorem 3.3. Every connected, simply connected c–step nilpotent Lie group G of dimension K embeds into P (RK ) in such a way that the image of G consists of polynomials of degree m for m =max(1,c 1) and so that the action of G on ≤ − RK is simply transitive. Proof. Write for the Lie algebra of G. It is well known that there is a one-to-one correspondence between and G via the exponential mapping. We follow Mal0cev’s ideas to describe the group structure of G. Consider the sequence of Lie subalgebras hi of given by the lower central c c 1 series of in reversed order. So h = (1) = , h − = (2) =[,], ... , 1 (c) (c 1) h = =[, − ]. Choose a basis
A1,1,A1,2,... ,A1,k1 ,A2,1,... ,A2,k2 ,A3,1 ... ,Ac,kc i of in such a way that h is the vector space generated by A1,1,A1,2,... ,Ai,ki . Now, introduce a system of coordinates on G by defining for all g of G its coordinate
c(g) to be the coordinate of log(g) with respect to the basis A1,1,... ,Ac,kc .
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Consider x (resp. y) G,withlogx= xi,j Ai,j (resp. log y = yi,j Ai,j ). The Campbell–Baker–Hausdorff∈ formula implies that P P xy =exp( xi,j Ai,j )exp( yi,j Ai,j ) X X 1 =exp (xi,j + yi,j)Ai,j + xp,qyr,s[Ap,q,Ar,s]+ . 2 ··· A close examination ofX the previous expressionX now shows that the (i, j)–th coor- dinate of the product xy satisfies:
c(xy)i,j = xi,j + yi,j + Pi,j (xi+1,1,... ,xc,kc ,yi+1,1,... ,yc,kc )
where Pi,j is a polynomial of total degree c i + 1 in all the variables xp,q and ≤ − yr,s (use the nilpotency of ). Moreover, when regarded as a polynomial in the xp,q–variables (resp. yr,s–variables) alone, Pi,j has degree c i. So, in terms of the coordinates introduced for G, the≤ product− is expressed by polynomial functions. We use these functions to define the embedding of G in P (RK ) as follows: ρ : G P (RK ):g ρ(g)with K → 7→ y R :(ρ(g)(y))i,j = c(g)i,j +yi,j +Pi,j (c(g)i+1,1,... ,c(g)c,kc ,yi+1,1,... ,yc,kc ) ∀ ∈ It is now easy to see that this ρ indeed defines an action of G on RK (in fact, it is just left translation in G!), which satisfies the criteria of the statement. Corollary 3.4. If G admits a lattice N, then we may consider the torsion-free central series N of N determined by the isolators of the lower central series of N. ∗ If a1,1,a1,2,... ,ac,kc denotes a set of generators of N which is compatible with N , we may choose the basis used in the theorem to be determined by ∗
A1,1 =loga1,1,A1,2 =loga1,2,... ,Ac,kc =logac,kc . Restricting the representation ρ : G P (RK )(based on the chosen basis of ) to → N, we see that we obtain a canonical representation of N into P (RK ).
In view of this corollary, Counterexample 2.6 is really disappointing. It looks as if the only way out, if one insists on the capability to iterate the set-up, is to go on with the general polynomial (without upper bound degree) situation. Remark 3.5. The proof of Theorem 3.3 can easily be adjusted to work with more general central series. Therefore, if G admits a lattice N, we can find a canonical type polynomial representation of N (of finite degree) with respect to any torsion- free central series N of N. ∗ For technical reasons we need the following (rather well known) lemma, the proof of which is left to the reader. Lemma 3.6. Let Γ Rk be any set of elements which span Rk as a vector space, ⊆ for some k N0.Letp(x1,x2,... ,xk,xk+1,... ,xn) be a polynomial in n variables for n k, with∈ real coefficients. Suppose ≥ p(x1 + z1,x2 +z2,... ,xk +zk,xk+1,... ,xn)=p(x1,x2,... ,xk,xk+1,... ,xn) n for all (z1,z2,... ,zk) Γ and all (x1,x2,... ,xn) R . ∈ ∈ Then p(x1,x2,... ,xk,xk+1,... ,xn) does not depend on the variables x1,x2, ... ,xk.
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Lemma 3.7. Suppose N and N are as before. For any canonical type polynomial representation of N with respect∗ to N we have ∗ Ki k Ni/Ni 1 Ki+1 k P (R , R ) − ∼= P (R , R ) for all i, 1 i
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K k P (R , R ) with the normal subgroup subgroup N1 of N, we apply Lyndon-Serre spectral sequence (the restriction–inflation exact sequence) to get isomorphisms:
i K k i K k N/N1 i K2 k H (N,P(R ,R )) ∼= H (N/N1,P(R ,R ) ) ∼= H (N/N1,P(R ,R )), where the second isomorphism is due to Lemma 3.7. Finally the last term is trivial by induction hypothesis. This completes the proof.
Corollary 3.9. Let E be as above and assume ϕ : E Aut(P (RK , Rk)) is any → representation, which is conjugated inside Aut(P (RK , Rk)), to a canonical type polynomial representation ρ : E P (RK ) Aut(P (RK , Rk)),then → → i K k Hϕ(E,P(R ,R )) = 0, i>0. ∀ Lemma 3.10. Let E be a group containing a torsion-free, finitely generated nilpo- tent group N as a normal subgroup of finite index. If ρ : E P (RK ) is a repre- sentation of E, restricting to a canonical type polynomial representation→ of N,with respect to a torsion-free central series N ,thenρitself is canonical, with respect to E ,whereE is given by: ∗ ∗ ∗ E : E0 = N0 =1 E1 =N1 Ec 1 =Nc 1 Ec =Nc Ec+1 = E. ∗ ⊂ ⊂···⊂ − − ⊂ ⊂ Proof. It is enough to show that, for e E, ρ(e) is such that ∈ ρ(e):RK R K −→ : x =(x1,1,x1,2,... ,x1,k ,x2,1,... ,xi,j ,... ,xc,k ) ρ(e)(x), 1 c 7→ e e e e with ρ(e)(x)=(P1,1(x),... ,Pi,j (x),... ,Pc,kc (x)), and Pi,j (x) is a polynomial de- pending only on the variables xp,q with p i. e ≥ First we show that Pi,j (x) does not depend on x1,q,ifi>1. Let z be any element of N1, then there exists a z0 N1 such that ∈ (3) ez = z0e. We will look at the effect of both sides of (3) on a general element x with coordinates
(x1,1,x1,2,... ,xc,kc ). K Remember that z (resp. z0) acts on R by some translation on the first k1 components, say by (z ,... ,z )(resp.(z ,... ,z )). On the one hand we 1,1 1,k1 10,1 10,k1 have that ez x = e(z x) e = (z1,1 + x1,1,... ,z1,k1 + x1,k1 ,x2,1,x2,2,... ,xc,kc ) e e =(P1,1(z1,1+x1,1,... ,xc,kc ),... ,Pc,kc (z1,1 + x1,1,... ,xc,kc )), while, on the other hand, we see that ez x =z0e x = z0 (ex) e e =(z10,1,... ,z10,k1 , 0.... ,0) + (P1,1(x),... ,Pc,kc (x)) e e e e =(P1,1(x)+z10,1,... ,P1,k1 + z10 ,k1 ,P2,1(x),... ,Pc,kc (x)).
These two computations result in i>1, z N1 : ∀ ∀ ∈ e Pi,j (z1,1 + x1,1,... ,z1,k1 + x1,k1 ,x2,1,... ,xc,kc ) e = Pi,j (x1,1,... ,x1,k1 ,x2,1,... ,xc,kc ).
k1 Since the set of all (z1,1,... ,z1,k1 ) forms a lattice of R , we conclude that the poly- e nomial Pi,j (x1,1,... ,xc,kc ) has to be independent of the coordinates x1,q (1 q k1). ≤ ≤
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The same technique can now be used to proceed this proof. One shows by e induction on p that Pi,j is independent of xp,q if i>p. Remark 3.11. We will call a filtration E as in the lemma above a torsion-free pseudo–central series of E.IfEis virtually∗ nilpotent as above, then, if we consider a canonical type representation we will always mean “canonical with respect to a torsion-free pseudo-central series of E”. We finish this section with the following lemma:
Lemma 3.12. Let E be a virtually abelian group containing a group Zk as a normal subgroup of finite index. Then a canonical type polynomial representation is an affine representation.
Proof. Let ρ : E P (Rk) be the canonical type polynomial representation. We → know that the abelian normal subgroup acts on Rk via translations: z x = z + x z Zk and x Rk. ∀ ∈ ∀ ∈ k 1 There is a homomorphism ϕ : E Aut (Z ) such that eze− = ϕ(e)z.Fixan → element e and assume ϕ(e)isgivenbyamatrix(αi,j )1 i,j k. We denote ρ(e)= ≤ ≤ (P1(x),P2(x),... ,Pk(x)), where the Pi(x) are some polynomials in k variables. For k any element x =(x1,x2,... ,xk)andanyz=(z1,z2,... ,zk) Z we have that ∈ ez x = (ϕ(e)z)ex e (x1 + z1,... ,xk +zk)=ϕ(e)z+(P1(x),... ,Pk(x))
⇓ Pi(x1 + z1,... ,xk +zk)=αi,1z1 + +αi,kzk + Pi(x1,... ,xk). ··· (1 i k) ≤ ≤ Now, consider the polynomial
Q(x)=Pi(x) αi,1x1 αi,kxk Pi(0). − −···− − The previous computation shows that Q(z)=0 z Zk. This implies that Q(x)= ∀ ∈ 0orthatPi(x) is a polynomial of degree 1. 4. Existence and uniqueness of polynomial manifolds InthissectionwemakeintensiveuseofthebasicfactsofthetheoryofSeifert Fiber Space constructions. We refer to [Lee83] and [CR77] for the full details concerning this concept. With a polynomial manifold of dimension n we mean a quotient manifold of the form E Rn where E is a group acting freely and properly discontinuously on Rn via \ a polynomial action ρ : E P (Rn). If the image of ρ consists only of polynomials → of degree s, we say that the manifold E Rn is of degree s. Whenever we use the term manifold,≤ we will mean a closed manifold.\ ≤ Theorem 4.1. (Existence) Let N be a torsion-free, finitely generated nilpotent group of rank K. For any torsion-free central series N , there exists a canoni- ∗ cal type polynomial representation ρ : N P (RK ). → K (Uniqueness) If ρ1,ρ2 are two such polynomial representations of N into P (R ) with respect to the same N , there exists a polynomial map p P (RK ) such that 1 ∗ K ∈ K ρ2 = p− ρ1 p. Consequently, the manifolds ρ1(N) R and ρ2(N) R are “poly- nomially◦ diffeomorphic◦ ”. \ \
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Proof. (Existence) The existence is already covered by Corollary 3.4. Nevertheless, it is useful to present an alternative proof here; in fact, this proof, as it is based on a cohomological argument, has the advantage of generalising to the virtual nilpotent situation and offering a set-up allowing us to treat the uniqueness. We proceed by induction on the nilpotency class c of N.IfNis abelian, then the existence is well known. Now, suppose that N is of class c>1 and the existence is guaranteed for lower nilpotency classes. Using the induction hypothesis, the group N/N1 can be furnished with a canonical type polynomial representation
K ρ¯ : N/N1 P(R 2 ). → k1 K2 k1 K2 We obtain an embedding i : N1 = Z P (R , R ) if we define i(z):R ∼ → → Rk1 :x z. We are looking for a map ρ making the following diagram commuta- tive: 7→
1 N1 N N/N1 1 → → → → i ρ ϕ ρ¯ ↓ ↓ ↓ × 1 P(RK2 , Rk1 ) P (A P) Aut Zk1 P (RK2 ) 1 → → o × → × → ∩ P (RK ) K k k K where P o(A P ) denotes P (R 2 , R 1 )o(Aut Z 1 P (R 2 )) and ϕ : N/N1 k1 × × → Aut Z =AutN1 denotes the morphism induced by the extension 1 N1 → → N N/N1 1. →The existence→ of such a map ρ is now guaranteed by the surjectiveness of δ in the long exact cohomology sequence (4) δ 1 K2 k1 k1 2 k1 0 H (N/N1,P(R ,R )/Z ) H (N/N1, Z ) 0 ···→ → → → →···
1kkK2 k1 2 K2 k1 H(N/N1,P(R ,R )) (Theorem 3.8) H (N/N1,P(R ,R ))
(Uniqueness) Again we proceed by induction on the nilpotency class c of N.For c= 1 the result is again well known. Indeed, two canonical type representations of a virtually abelian group are even known to be affinely conjugated. So we suppose that N is of class c>1 and that the theorem holds for smaller nilpotency classes. The representations ρ1,ρ2 induce two canonical type polynomial representations K ρ¯1, ρ¯2 : N/N1 R 2 . → By the induction hypothesis, there exists a polynomial mapq ¯ : RK2 RK2 such that → 1 ρ¯2 =¯q− ρ¯1 q.¯ ◦ ◦ Lift thisq ¯ to a polynomial map q of RK1 as follows: x Rk1 , y RK2 : q : RK1 RK1 :(x, y) q(x, y)=(x, q¯(y)). ∀ ∈ ∀ ∈ → 7→ When restricted to N1, ρ1 and ρ2 are mapping elements onto pure translations of Rk1 . We know that there exists an affine mapping A˜ : Rk1 Rk1 for which → ˜ 1 ˜ n N1 : ρ2 k = A− ρ1 k A. ∀ ∈ |R 1 ◦ |R 1 ◦
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Extend A˜ to an affine mapping of RK by defining x Rk1 , y RK2 : A : RK1 RK1 :(x, y) A(x, y)=(A˜(x),y). ∀ ∈ ∀ ∈ → 7→ 1 1 Let us denote ψ = A− q− ρ1 q A.Thenweseethatψand ρ2 are two canonical type polynomial representations◦ ◦ ◦ ◦ of N, which coincide with each other on N1 and which induce the same representation of N/N1. This means that ψ and ρ2 can be seen as the result of a Seifert construction with respect to the same data (cf. the commutative diagram above). Now we use the injectiveness of δ in (4), which implies that ψ and ρ2 are conju- gated to each other by an element r P (RK2 , Rk1 ) (seen as an element of P (RK )!). So we may conclude that ∈ 1 1 1 1 1 ρ2 = r− ψ r = r− A− q− ρ1 q A r = p− ρ1 p ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ if we take p = q A r. ◦ ◦ Corollary 4.2. Any canonical polynomial representation with respect to some tor- sion–free central series of N is polynomially conjugated to a canonical type polyno- mial representation of N of degree s,wheres=max(1,c 1). Consequently, for each N there is a well defined number≤ − m(N)= Min s N N is fundamental group of a polynomial manifold of degree s . { ∈ || ≤ } Corollary 4.3. Since any canonical type polynomial representation ρ of N is poly- nomially conjugated to a polynomial representation of finite degree, ρ itself has to be of finite degree. By replacing N by a group E which contains N as a normal subgroup of finite index, and noting that i res i H (E,P(RK,Rk)) H (N,P(RK,Rk)) → is injective, we get the following Corollary 4.4. Let E be a group containing a normal subgroup N, which is torsion- free, finitely generated of rank K, nilpotent of class c and of finite index in E and fix a torsion-free pseudo–central series E of E. Then there exists a canonical type polynomial representation of E with respect∗ to this series. Moreover, if ρ1,ρ2 are two canonical type polynomial representations of E into P (RK ), with respect to E , then there exists a polynomial map p P (RK ) such that 1 ∗ ∈ K ρ2 = p− ρ1 p.IncaseEis torsion-free this means that the manifolds ρ1(E) R ◦ K◦ \ and ρ2(E) R are “polynomially diffeomorphic”. \ Corollary 4.5. Suppose E acts properly discontinuously on RK ,viaϕ:E → Aff(RK ) and suppose also that there exists a canonical type affine representation ρ : E Aff(RK ),then → i K k Hϕ(E,P(R ,R )) = 0, i>0. ∀ Proof. The results of [FG83] imply that ϕ and ρ are conjugated inside P (RK ). Hence Corollary 3.9 can be applied. Corollary 4.6. Suppose E is a virtually 3-step nilpotent group of rank K,acting properly discontinuously on RK ,viaϕ:E Aff(RK ),then → i K k Hϕ(E,P(R ,R )) = 0, i>0. ∀
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Proof. We can apply the previous corollary, since any virtually 3-step nilpotent group admits a canonical type affine representation ([Lee83]).
Remark 4.7. Recently Hartl ([Har91]) proved that for N finitely generated, torsion- freenilpotent of class c, H2(N,Zk) is polynomial of degree c. It, once again, ≤ K k looks tempting to investigate its relation with working in Pc(R , R ). However, Counterexample 2.6 once again obstructs an elaboration of this approach.
In view of Milnor’s problem mentioned in the beginning and recent counterex- amples obtained by Benoist ([Ben92]) and Burde and Grunewald ([BG93]), it looks interesting to introduce the following concept.
Definition 4.8. For N finitely generated, torsion-free nilpotent of class c and rank K, we define the affine defect of N, d(N), to be
d(N)=m(N) 1. − Lemma 4.9. For N as above, we can say that 1. d(N)=0iff N is fundamental group of a complete affinely flat compact manifold. 2. 0 d(N)=m(N) 1 Max 0,c 2 . This follows at once from the results above.≤ − ≤ { − } 3. If c 3 or K 7,thend(N)=0. The first is proved in [Sch74]; for the second,≤ we refer≤ to [Ben92].
The affine defect d(N) measures the obstruction for N to be fundamental group of a complete affinely flat compact manifold. As we will show in our last section, the counter-examples to Milnor’s question studied so far are of weakest possible affine defect.
Remark 4.10. Corollary 4.4 is a generalization of Theorem 1.20 in [FG83] and of Theorem 4.1 in [IL90]. In this second paper, the authors are concerned with canon- ical type affine representations (called “special” there), with respect to the upper central series. They claim that we can take the polynomial to conjugate with, of degree c!. Certainly there is a polynomial conjugation, but the degree does not seem to≤ be right. We indicate a gap in the proof of Theorem 4.1 of [IL90].
First notice that, in the case where c = 1, the polynomial needed to conjugate two canonical type affine representations should be taken of degree 1. The real gap is in the induction argument in [IL90]. Indeed, an induction argument assumes that, after conjugation by a polynomial on the N/N1 = N/Z(N)-level, the result on the N–level is again a canonical type affine representation. Of course, the conjugation does not touch the canonical type character of the representation. However, as the subsequent example shows, one might obtain a polynomial representation which stays outside the affine group.
Example 4.11. Let N be the following finitely generated torsion-free 3–step nilpo- tent group:
N : a1,a2,b,c [a2,a1]=b, [b, a1]=c, [b, a2]=[c, a1]=[c, a2]=[c, b]=1. h k i
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We propose the following two canonical type affine representations of N (upper central series), where we indicate only the images of the generators: 1 10 1 0 1000 0 010− 1 0 0100 0 ρ (a )= , ρ(a)= , 1 1 0010−112 00100 0001 0 0001 1 1000 0 1000 1 0100 1 0100 0 ρ(b)= , ρ(c)= , 1 001001 00100 0001 0 0001 0 and 2 1 3 00 0 1000 0 010− 1 0 01 1 0 0 ρ (a )= 2 , ρ(a)= 2 , 2 1 0010−122 00100 0001 0 0001 1 1 2 10 3 0 3 1000 1 0100 1 0100 0 ρ(b)= , ρ(c)= , . 2 001002 00100 0001 0 0001 0 The presentation for N/Z(N) can be read from the presentation ofN,bysimply replacing all c’s by 1. The induced canonical type affine representationsρ ¯1, ρ¯2 : N/Z(N) Aff(R3) are also easy to write down: → In the linear part of ρi (i =1,2), we discard the first column and the first • row. In the translational part, we omit the first term. • Following the general theory of Seifert Fiber Spaces and the proof of Theorem 4.1 in [IL90] or of Theorem 4.4 we find that there must exist a polynomial mapping p(x)inP(R2,R) such that 1 ρ¯2 = p− ρ¯1 p ◦ ◦ when p(x) is seen as an element of P (R2, R)oP (R2)orasanelementofP(R3). Some elementary computations show that the only possible p’s are of the form: z z xy + r − 2 p : R3 R3 : y y → x 7→ x where r can be any real number. If we now lift this p to a polynomial mapping of R4,say t t z z xy + r q:R4 R4: − 2 → y 7→ y x x we find that xy t t + x z + 2 r − x − 1 4 4 z z 2 q− ρ1(a1)q : R R : − → y 7→ y +1 x x
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which is clearly not affine. Moreover, the same can be said about any lift of p of the form t m(x, y, z, t) z z xy + r q : R4 R4 : − 2 → y 7→ y x x where m(x, y, z, t) denotes some polynomial of degree 2 in the four variables. ≤ Finally, we indicate how ρ1 and ρ2 can be seen to be conjugated by a polynomial. Consider t xy xy2 2z yz t + 3 + 6 3 3 z z xy− − p : R4 R4 : 2 . → y 7→ −y x x Then one can see that 2z yz t t + 3 + 3 xy 1 4 4 z z + 2 p− : R R : → y 7→ y x x and some calculations then result in
1 ρ2(n)=p− ρ1(n) p n N. ◦ ◦ ∀ ∈
5. Groups of affine defect one The only known connected, simply connected nilpotent Lie groups not acting simply transitively and affinely on Euclidean n-space are those constructed by Benoist [Ben92] and Burde and Grunewald [BG93]. In fact, it is the situation on the Lie algebra level which is investigated in both papers. The Lie algebras in question are 11-dimensional; a particularly interesting family of them is denoted by a( 2,s,t) and has a basis e1,e2,... ,e11. Moreover, the brackets of these Lie algebras− are completely determined by
[e1,ei]=ei+1, 2 i 10, ≤ ≤ [e1,e11]=0, [e2,e3]=e5 and [e2,e5]= 2e7+se8 + te9. − The results of Benoist, Burde and Grunewald imply that, in case s =0,a( 2,s,t) has no faithful 12-dimensional linear representation. Therefore, no6 lattice− of the corresponding Lie group ( 2,s,t) can act properly discontinuously on R11,via affine motions and with compactA − quotient. In this section we indicate how one can construct a uniform lattice Γ of ( 2,s,t), A − acting properly discontinuously on R11 via polynomial diffeomorphisms, in such a way that the polynomial mappings used are of degree 2. It will follow that these lattices Γ have affine defect 1. ≤ To do so, we will embed this Γ into a connected, simply connected Lie group P (of finite dimension) of polynomial diffeomorphisms of R11.Infact,letPbe the
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group consisting of all mappings
y11 y11 + p11(y1,y2,... ,y10) y10 y10 + p10(y1,y2,... ,y9) 11 11 . . f : R R : . . → 7→ y2 y2 +p2(y1) y1 y1 +p1 where pi(y1,y2,... ,yi 1) is a polynomial of degree 1inthevariablesy1,... ,yi 1, − ≤ − except p11 which is of degree 2. In view of the results mentioned≤ above, an embedding Γ , P is really the best (non affine) thing we could expect. → The first step to clear this job was to construct a unitriangular matrix repre- sentation of ( 2,s,t). This was not difficult since Burde and Grunewald [BG93] give an explicitA − way to construct a 34–dimensional (and with some more trouble, a 22–dimensional) matrix representation for the Lie algebra a( 2,s,t), using its universal enveloping algebra. These matrices are upper triangular,− with 0’s on the diagonal, and so by exponentiating we find the desired representation on the Lie group level. For the sake of simplicity we will restrict ourselves to the cases where s, t Z; however, we remark here that things work out for rational s and t too. ∈ Let Γ denote the subgroup of ( 2,s,t) generated by A1,A2,... ,A11,where A − A1 =exp(e1) A2 =exp(e2) A3 =exp(e3) 1 1 1 A4 =exp(2e4) A5 =exp(6e5) A6 =exp(24 e6) 1 1 1 . A7 =exp(120 e7) A8 =exp(720 e8) A9 =exp(25200 e9) 1 1 A10 =exp(1008000 e10) A11 =exp(90720000 e11) Making use of the matrix representation of ( 2,s,t), and so of Γ, mentioned above, one checks that Γ is a uniform lattice ofA −( 2,s,t)with A − 1 2 5 109 4(79 60s) 6( 1108 525s 1400t) Γ= A1,... ,A11 [A2,A1]= A3− A4A5A6− A7 A8 − A9 − − − h k 2267555+151088s 126000t .A10− − 2(203230225 12930435s+677376s2 +4372200t) .A11 − 2 3 4 185 6( 91 60s) 35(829+180s 360t) [A3,A1]= A4− A5A6− A7 A8 − − A9 − 80(45515 3066s+3150t) .A10 − 180( 5202055+135870s+18816s2 190050t) .A11 − − 6 120 360(3+s) 840( 26 15s+15t) [A3,A2]= A5− A7− A8 A9 − − 6720( 585 28s 75t) 3780( 41975 13035s 2688s2 +5150t) i .A10 − − − A11 − − − 3 6 10 15 16905 140( 2425 864s) [A4,A1]= A5− A6A7− A8 A9 A10 − − 90(836225+60480s+28224s2 222075t) .A11 − , 12 900 12600s 8400(341+60t) 13494600s [A4,A2]=A6− A8− A9 A10 A11 , 180 360s 12600t 5896800 85050(4725+334s) [A4,A3]=A7− A8 A9 A10− A11− , 4 10 20 175 1400 21708750 [A5,A1]=A6− A7 A8− A9 A10− A11 , 40 120s 840(13 5t) 94080s 1260( 12350+4032s2 7725t) [A5,A2]=A7 A8− A9 − A10 A11 − − ,
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360 4200s 168000t 91003500 Γ–continued :[A5,A3]=A8− A9 A10 A11 , 2 11340 80640s 3780( 448s +3525t) [A5,A4]=A9− A10 A11 − , 5 15 175 1750 31500 [A6,A1]=A7− A8 A9− A10 A11− , 150 2100s 42000( 13 2t) 6095250s [A6,A2]=A8 A−9 A10 − − A11− , 2 2520 1680s 1890(448s 1525t) [A6,A3]=A9 A10 A11 − , 113400 1814400s [A6,A4]=A10− A11 , 10843875 [A6,A5]=A11− , 6 105 1400 31500 [A7,A1]=A8− A9 A10− A11 , 2 546 17136s 189( 6175 896s 4950t) [A7,A2]=A9 A10− A11 − − − , 65520 695520s [A7,A3]=A10 A11− , 4465125 [A7,A4]=A11 , 35 700 21000 [A8,A1]=A9− A10 A11− , 7280 141120s [A8,A2]=A10− A11− , 505575 [A8,A3]=A11− , 40 1800 [A9,A1]=A10− A11 , 4275 [A9,A2]=A11− , 90 [A10,A1]=A11− , [Ai,Aj]=0for1 j11 . ≤ ≤ i
In fact, after having taken A1 =exp(e1)andA2 =exp(e2), the others have been chosen carefully so that they generate a lattice. It is obvious that this choice is not unique. Now that we have enough information about the group structure, we are go- ing to build up an embedding Γ P . In order to do so we will first realise → the rank 10 group Γ/Z (Γ) = Γ/ A11 as a properly discontinuous group of affine h i motions of R10. The idea behind this is the fact that the adjoint representation ad : a( 2,s,t) gl(a( 2,s,t)) is an 11-dimensional matrix representation factor- − → − ing through a( 2,s,t)/Z (a( 2,s,t)) = a( 2,s,t)/ e11 . By exponentiating this representation,− we may realise− ( 2,s,t)/Z−( ( 2,s,th ))i as a group of affine mo- A − A − tions of R10. However, in this way the group ( 2,s,t)/Z ( ( 2,s,t)) does not A − A − act simply transitively on R10 as we would like, therefore we slightly modify the adjoint representation in order to obtain, on the Lie group level, a simply transitive action. This altered adjoint representation, denoted by φ, is determined completely by the images of e1 and e2:
0100000000 0 0010000000 0 0001000000 0 0000100000 0 0000010000 0 φ(e1)= 0000001000 0 0000000100 0 0000000010 0 0000000000 0 0000000000 1 − 0000000000 0
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and
19 28 s 448 s2+2475 t 00 16 25 2000 000000 26 51 s 00 0 5 25 2 t 00000 13 00 0 0 5 2 st000 0 00 0 0− 0 5 s 000 0 00 0 0 0− 0 2000 0 − φ(e2)= 00 0 0 0 0 0 100 0. 00 0 0 0 0 0 010 0 00 0 0 0 0 0 000 0 00 0 0 0 0 0 000 1 00 0 0 0 0 0 000− 0 00 0 0 0 0 0 000 0 Let us define the morphism ϕ as the composite map:
log φ exp ( 2,s,t) a ( 2 ,s,t) a ( 2 ,s,t)/ e11 gl(11, R) Gl(11, R). A − −→ − −→ − h i −→ −→ As the images of ϕ consist of unitriangular matrices, ( 2,s,t)/Z ( ( 2,s,t)) can A − A − indeed be looked at as being a simply transitive group of affine motions of R10. In the next step we lift the map ϕ :Γ Gl(11, R) to a faithful morphism ψ :Γ P.Wepresentψby giving the images→ of the generators: →
y11 y11 + fi(y1,y2,... ,y10) y10 y10 11 11 . . ψ(Ai):R R : . . . → 7→ ϕ(Ai) y2 y2 y1 y1 10 Here, we see ϕ(Ai) as an affine mapping of R and the fi are polynomials of degree 2 given by: ≤ 99569925 t + 17013024 s2 250 1 1 f1(y1,... ,y10)= − y2 y3 y4 90720000 − 40320 − 5040 1 1 1 1 1 y5 y6 y7 y8 y9 y10 −720 − 120 − 24 − 6 − 2 − 5458475 + 7392820 s + 1937376 s2 + 10623200 t +− y2y3 1680000 142755 6860 s + 3136 s2 38675 t +− − − y2y4 84000 1617 320 s 960 t 800 40 s 448 s2 + 1525 t + − − y2y5 + − − − y2y6 1920 2000 195 + 46 s 321 +− y2y7 + y2y8 50 80 38025 2560 s + 1792 s2 14100 t + − − y3y4 16000
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15075 1920 s + 896 s2 7050 t 3(45 16 s) + − − y3y5 + − y3y6 4000 50 189 1377 1377 y3y7 + y4y5 + y4y6 − 16 160 80 6486150 22287100 s 5727456 s2 1128960 s3 32661825 t − − − − y2 − 10080000 2 11881800 st 627615 13440 s + 12544 s2 98700 t + y2 + − − − y2 10080000 2 336000 3 459 + y2 160 4
69500 3263904 s2 376320 s3 + 6044325 t 3960600 st f2(y1,... ,y10)=− − − − y3 3360000 21465 s 6272 s3 66010 st + − − y4 28000 8843500 2685984 s2 + 7797825 t 198 s + − − y + y 1680000 5 875 6 15775 6272 s2 34650 t 28 s 19 + − − y7 + − y8 + − y9 28000 25 16 767800 + 210464 s2 120825 t + − − y2y3 160000 433649 s + 6272 s3 + 66010 st + − y2y4 28000 519000 + 83104 s2 218825 t 517 s 4089 + − y2y5 + y2y6 y2y7 40000 14000 − 560 11391300 + 309792 s2 + 2217775 t 1671 s + − y3y4 + y3y5 560000 350 21087 5913 + y3y6 y4y5 1400 − 560 16554375 339872 s + 150528 s3 + 1584240 st + − − y2 1344000 2 1671 s 33039 s + y2 + − y2 700 3 2800 4 3 (2187 3584 s) 62901 448 s2 1525 t 23 s f3(y1,... ,y10)= − y4 y5 + − y6 + − y7 2800 − 2800 2000 25 321 41175 6272 s3 66010 st y8 + − − − y2y3 − 80 28000 11 9184 s2 + 29075 t 19407 s 549 + − y2y4 + − y2y5 y2y6 80000 14000 − 560 2 3573 s 76383 26336 s 457825 t 2 2673 2 + y3y4 + y3y5 + − − y y , 400 1400 80000 3 − 112 4 448 s2 + 3525 t 12 s 189 26336 s2 + 457825 t f4(y1,... ,y10)=− y5 + y6 + y7 + y2y3 4000 25 32 160000 3 123183 s 35397 6561 448 s + 4715 st 2 y2y4 y2y5 + y3y4 + y , − 28000 − 1120 1400 4000 2
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2 3157 s 2 12793 t 2 39 s f5(y1,... ,y10)= y y y2y3 15000 2 − 19200 2 − 1750 2187 2 891 459 y + y2y4 y6, −1400 3 224 − 160 6469 s 8073 f (y ,... ,y )= y2 + y y , 6 1 10 112000 2 22400 2 3 183 f (y ,... ,y )= y2, 7 1 10 22400 2 f8(y1,... ,y10)=0,
f9(y1,... ,y10)=0,
f10(y1,... ,y10)=0, 1 f(y ,... ,y )= . 11 1 10 90720000 Theorem 5.1. Let ( 2,s,t) be the Lie group which stands in one-to-one corre- A − spondence with the Lie algebra a( 2,s,t).Ifs, z Z,then ( 2,s,t) acts simply − ∈ A− transitively and via polynomial diffeomorphisms of degree 2 on R11.So,ifs=0, then any lattice of ( 2,s,t) is a group of affine defect one.≤ 6 A − Proof. By the computations above we know that ( 2,s,t) contains a lattice Γ which embeds nicely into a Lie group P of polynomialA − diffeomorphisms of degree 2. Observe that this group P , introduced above, is nilpotent. This can be seen ≤ as follows. Let P1 be the normal subgroup of P consisting of those polynomial diffeomorphisms for which pi 0(1 i 10). So P1 consists of those polynomials 11 ≡ ≤ ≤ of R which only affect 1 coordinate. Denote by P2 the subgroup of P determined by those polynomials for which p11 0. So P2 is isomorphic to the group of unitriangular 11 11–matrices. There≡ is a split short exact sequence × 1 P1 P P2 1. → → → → P2 is a nilpotent group, and so γn(P ) P1 if n is big enough. It is now an exercise to see that further commutator subgroups⊆ will gradually get smaller and smaller until, at the end, they vanish. Using [Gor71, Proposition 2.5], the embedding Γ P extends uniquely to an embedding of the Lie group ( 2,s,t) P, making→ it acting properly discontin- A − → uously on R11, which was to be shown.
Let us point out finally that the use of MATHEMATICAr [Wol93] was quite crucial in setting up and performing the computations above. A set of procedures enabling the reader to verify these results is available on request from the authors.
References
[Ben92] Yves Benoist, Une nilvari´et´e non affine,C.R.Acad.Sci.ParisS´er. I Math. 315 (1992), pp. 983–986. MR 93j:22008 [Ben95] Yves Benoist, Une nilvari´et´e non affine, J. Differential Geom. 41 (1995), pp. 21–52. [BG93] Dietrich Burde and Fritz Grunewald, Modules for certain lie algebras of maximal class, Preprint (1993), to appear in J. Pure and Applied Algebra. [Boy89] Nguiffo B. Boyom, The lifting problem for affine structures in nilpotent lie groups,Trans. Amer. Math. Soc. (1989), pp. 347–379. MR 89i:53035 [Boy90] Nguiffo B. Boyom, Sur les structures affines homotopesaz` ` ero des groupes de lie,J. Differential Geom. (1990), pp. 859–911. MR 91g:53026
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use POLYNOMIAL STRUCTURES FOR NILPOTENT GROUPS 97
[CR77] Pierre E. Conner and Frank Raymond, Deforming homotopy equivalences to homeomor- phisms in aspherical manifolds, Bull. A.M.S. 83 (1977), pp. 36–85. MR 57:7629 [DI93] Karel Dekimpe and Paul Igodt, Computational aspects of affine representations for tor- sion free nilpotent groups via the seifert construction, J. Pure Applied Algebra (1993), pp. 165–190. MR 94e:20049 [DI94] Karel Dekimpe and Paul Igodt, The structure and topological meaning of almost–torsion free groups, Comm. Algebra 22 (1994), pp. 2547–2558. MR 95f:20057 [FG83] David Fried and William M. Goldman, Three–dimensional affine crystallographic groups, Adv. in Math. 47 (1983), pp. 1–49. MR 84d:20047 [Gor71] V. V. Gorbaceviˇc, Discrete subgroups of solvable Lie groups of type (E), Math. USSR Sbornik 14 (1971), pp. 233–251. MR 36:6448 [Har91] Manfred Hartl, Abelsche Modelle Nilpotenter Gruppen, Inaugural-Dissertation RFW Univ. Bonn, 1991. [IL90] Paul Igodt and Kyung Bai Lee, On the uniqueness of certain affinely flat infra- nilmanifolds, Math. Zeitschrift 204 (1990), pp. 605–613. MR 91e:57066 [Lee83] Kyung Bai Lee, Aspherical manifolds with virtually 3–step nilpotent fundamental group, Amer. J. Math. 105 (1983), pp. 1435–1453. MR 85d:57032 [LR81] Kyung Bai Lee and Frank Raymond, Topological, affine and isometric actions on flat riemannian manifolds, J. Differential Geometry 16 (1981), pp. 255–269. MR 84k:57027 [LR84] Kyung Bai Lee and Frank Raymond, Geometric realization of group extensions by the seifert construction, Contemporary Math. A. M. S. 33 (1984), pp. 353–411. MR 86h:57043 [Mal51] AnatoliˇıI.Mal0cev, On a class of homogeneous spaces, Translations A.M.S. (1951), no. 39, pp. 1–33. MR 12:589e [Mil77] John Milnor, On fundamental groups of complete affinely flat manifolds, Adv. Math. 25 (1977), pp. 178–187. MR 56:13130 [Nis90] Mounir Nisse, Structure affine des infranilvari´et´es et infrasolvari´et´es, C. R. Acad. Sci. Paris S´erie I 310 (1990), pp. 667–670. MR 91e:57069 [Sch74] John Scheuneman, Affine structures on three-step nilpotent lie algebras,Proc.A.M.S.46 (1974), pp. 451–454. MR 54:470 [Seg83] Daniel Segal, Polycyclic Groups, Cambridge University Press, 1983. MR 85h:20003 [Wol93] Stephen Wolfram, Mathematica, Wolfram Research, Inc., 1993.
Katholieke Universiteit Leuven, Campus Kortrijk, B-8500 Kortrijk, Belgium E-mail address: [email protected] Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019 E-mail address: [email protected]
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